Question about Covariance matrix from 'SPA' function

Question

Achille Nicoletti 2016-2-7

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https://ww2.mathworks.cn/matlabcentral/answers/267037-question-about-covariance-matrix-from-spa-function

回答： Mayank Dwivedi 2016-2-9

Hello MATLAB geniuses,

So when one identifies the frequency response of a system using the SPA function, we can obtain the covariance matrix. Suppose I have an input 'u' and output 'y' of an experiment, with 'Ts' being the sampling time, 'winSize' being the window size of the identification, and 'freq' being the vector of frequency points to use in obtaining the frequency response. Then the covariance matrix can be obtained as follows:

dat = iddata(y,u,Ts)
G = spa(data,winSize,freq)
Covariance_Matrix = squeeze(G.CovarianceData)

For my case, the covariance matrix (at each frequency) is a 2x2 matrix, where Covariance_Matrix(k,1,1) is the variance in the real part, and Covariance_Matrix(k,2,2) is the variance in the imaginary part (for any positive integer k). My question is, how exactly are these values calculated? Do these values depend on the output noise spectrum? Basically, if I wanted to calculate this covariance matrix myself, how would I do it? The documentation, unfortunately, does not seem to mention anything about this.

Your help is much appreciated!

-Achille